Defects in Graphene: QFT approach

Theory

The derivation of the formalism used in this package can be found here.

For a more detailed discussion, see https://doi.org/10.1103/PhysRevB.102.195416

Use

Basics

This package can calculate the electronic density at impurity states and graphene atoms, see here.

The general flow goes as follows:

1. Create a new system by calling new_graphene_system() and assigning it to a variable, say my_system. This produces a pristine graphene monolayer with μ = 0, T = 0, no defects, and no modified atoms.
2. If desired, one can create impurity states by calling new_impurity(E::Float64), where E is the state energy, and assigning the output to a variable.
3. Once the impurities are created, it is possible to couple them to graphene sites using add_coupling(imp::ImpurityState, V::Float64, coord::GrapheneCoord), where V is the coupling strength. For example, we want to connect imp1 to a graphene site of the A sublattice at the unit cell (3,4). This is accomplished as follows: imp1 = add_coupling(imp1, V, graphene_A(3,4)). The procedure can be repeated to connect the impurity to more graphene sites. Couplings can be removed by calling imp1 = remove_coupling(imp1, n), where n is the index of the coupling in the array.
4. Next, the impurities need to be included in my_system by calling my_system = add_imp(my_system, imp1).
5. In a similar fashion, one can create perturbed atoms by calling perturbed_atom = perturbed_atom(E::Float64, dt::Float64, coord::GrapheneCoord), where E is the new on-site energy, dt is the modification of the hopping parameter with the atom's neighbors, and coord is the atom's location. Just as with the impurities, the atoms must be added to the system by calling my_system = add_mod_atom(my_system, perturbed_atom)
6. It is possible to remove impurities or perturbed atoms by calling my_system = remove_imp(my_system, n) or my_system = remove_mod_atom(my_system, n), where n is the index of the impurity/atom in the array.
7. The chemical potential and the temperature of the system can be changed by calling my_system = set_μ(my_system, new_mu) or my_system = set_T(my_system, new_T).
8. Finally, one can calculate the charge densities. To get the defect-induced charge density in graphene, call δρ_Graphene(coord, my_system), where coord is the coordinate of the atom. For the impurities, call ρ_All(my_system) if all the occupation numbers are desired, or ρ_Impurity(n, my_system) to only get the occupation number for the n'th impurity.
9. The spectral function at energy ω can be obtained by calling spectral_graphene(ω, coord, my_system) or spectral_impurity(ω, n, my_system)

Scripts

To facilitate the use of the package, a few useful files can be found in the calculation folder.

Density and Spectral Function Maps

To take advantage of parallel computation when computing the induced charge density over many sites

1. Start by defining the system in the computations_settings.jl file. This includes creating a new_graphene_system() and populating it with defects, as outlined above
2. In the same file, define the size of the system by specifying how many unit vectors it spans by setting nPts to a desired value. This means that the lattice will go from -nPts to +nPts for each of the unit vectors so that the total number of points on the lattice to be computed will then be 2 x (2nPts+1) x (2nPts+1). The "2" in the front originates from the two sublattices.
3. In rho_calculation.jl (for the density) or spectral_calculation.jl (for the spectral function), define the number of workers to be used in the parallel calculation by setting procs to a desired value.
4. Make sure that the targed directory to save the output exists and run the script! The induced density will be calculated at every lattice point and saved to two files, one for each sublattice.

Atom-resolved Spectral function

Analysis and Plotting

Plotting

When one performs the lattice-wide calculations, as described above, the output is saved to two files. These files only contain the values at the lattice sites without the atom coordinates. To plot these results, further processing is necessary. A function called Data_Process(lattice_A_data, lattice_B_data) in src/analysis/dataprocess.jl takes the data for the two sublattices and combines it by matching the data to the appropriate spatial coordinate, making it possible to plot the results. For more detail, see calculation/lattice_Plotter.jl.

Fourier Transform

ToDo

1. Implement the energy calculation
2. Fix up the Fourier Transform part